Systems and method for use of single mass flywheel alongside torsional vibration damper assembly for single acting reciprocating pump

ABSTRACT

A pump system may include a pump, a driveshaft, driving equipment, and a vibration dampening assembly configured to reduce pump-imposed high frequency/low amplitude and low frequency/high amplitude torsional vibrations. The pump may have an input shaft connected to the driveshaft. The driving equipment may include an output shaft having an output flange connected to the driveshaft. The driving equipment may be configured to rotate the driveshaft to rotate the input shaft of the pump therewith. The vibration dampening assembly may include one or more flywheels operably connected to the input shaft and configured to rotate therewith.

PRIORITY CLAIMS

This is a continuation of U.S. Non-Provisional application Ser. No.17/363,151, filed Jun. 30, 2021, titled “SYSTEMS AND METHOD FOR USE OFSINGLE MASS FLYWHEEL ALONGSIDE TORSIONAL VIBRATION DAMPER ASSEMBLY FORSINGLE ACTING RECIPROCATING PUMP,” which is a continuation of U.S.Non-Provisional application Ser. No. 17/213,562, filed Mar. 26, 2021,titled “SYSTEMS AND METHOD FOR USE OF SINGLE MASS FLYWHEEL ALONGSIDETORSIONAL VIBRATION DAMPER ASSEMBLY FOR SINGLE ACTING RECIPROCATINGPUMP,” now U.S. Pat. No. 11,092,152, issued Aug. 17, 2021, which is acontinuation of U.S. Non-Provisional application Ser. No. 16/948,291,filed Sep. 11, 2020, titled “SYSTEMS AND METHOD FOR USE OF SINGLE MASSFLYWHEEL ALONGSIDE TORSIONAL VIBRATION DAMPER ASSEMBLY FOR SINGLE ACTINGRECIPROCATING PUMP,” now U.S. Pat. No. 11,015,594, issued May 25, 2021,which claims priority to and the benefit of U.S. Provisional ApplicationNo. 62/704,560, filed May 15, 2020, titled “SYSTEMS AND METHOD FOR USEOF SINGLE MASS FLYWHEEL ALONGSIDE TORSIONAL VIBRATION DAMPER ASSEMBLYFOR SINGLE ACTING RECIPROCATING PUMP,” and U.S. Provisional ApplicationNo. 62/899,963, filed Sep. 13, 2019, titled “USE OF SINGLE MASS FLYWHEELALONGSIDE TORSIONAL VIBRATION DAMPER SYSTEM FOR SINGLE ACTINGRECIPROCATING PUMP,” the disclosures of which are incorporated herein byreference in their entireties.

BACKGROUND Technical Field

The present disclosure relates to single acting reciprocating pumps and,more specifically, to single mass flywheels and torsional vibrationdampers for use with single acting reciprocating pumps.

Discussion of Related Art

During fracturing operations, high and low frequency torsional vibrationis a common occurrence through the driveline. Such torsional vibrationis typically generated via the operation of a reciprocating pump.Reciprocating pumps are driven to pump “slugs” of fluid with as the pumpreciprocates or cycles. The speed and operating pressure of the pumpinfluences the amount of fluid pumped downstream of the pump. As thereciprocating pump is cycled, movement of the slugs create pressurefluctuations within fluid downstream of the pump. This pressurefluctuation may create “hydraulic fluid pulsation” within the pump thatis added to the operating pressure of the pump. The hydraulic fluidpulsation may be transferred upstream to driving equipment used to drivethe pump in the form of torque output variances. The driving equipmentmay include one or more components including, but not limited to, adriveshaft, an engine, a transmission, or a gearbox.

As noted, the nature of the suction and discharge strokes of thereciprocating pump generate variable torque spikes that originate fromthe discharge of high pressure fluid and may migrate through the driveline and cause damage and premature wear on the driveline componentsincluding the prime mover. Problematically, each reciprocating pumpsoperating in the field generally have their own torsional vibrationfrequency and amplitude profile that is dependent upon the selectedoperational pressure and rate. Another problem arises when a group ofreciprocating pumps are connected to a common discharge line. In thisoperational scenario, reciprocating pumps may begin to synchronize suchthat the natural sinusoidal wave form of one pump will begin to mirrorthat of another pump from the group, which promotes pressure spikes andtorsional distortion of even higher amplitude to pulsate through thedrive lines.

The torque output variances may create shock loading in the pump and inthe driving equipment upstream from the pump. This shock loading mayshorten the life of the driving equipment including causing failure ofone or more components of the driving equipment. In addition, drivingequipment such as combustion engines, e.g., gas turbine engines, have amovement of inertia, natural damping effects, and stiffnesscoefficients. Some driving equipment may have low natural dampingeffects that may allow for torsional resonance interaction within thedriving equipment and/or between the driving equipment and the pump.This torsional resonance may shorten the life of the driving equipmentincluding causing failure of one or more components of the drivingequipment.

Thus there is a need to provide protection of hydraulic drive linefracturing equipment from imposed high frequency/low amplitude and lowfrequency/high amplitude torsional vibrations.

SUMMARY

This disclosure relates generally to vibration dampening assemblies foruse with pump systems including a reciprocating pump and drivingequipment configured to cycle the pump. The vibration dampeningassemblies may include single mass flywheel(s) and/or torsionalvibration dampener(s) to reduce or eliminate upstream shock loadingand/or dampen torsional resonance from reaching the driving equipment;i.e., to reduce or eliminate pump imposed high frequency/low amplitudeand low frequency/high amplitude torsional vibrations.

According to some embodiments, a single mass flywheel or a series ofsingle mass flywheels along the drive-train system components betweenthe gear box or transmission and input shaft of a reciprocating pump maybe used to reduce output speed fluctuations that may cause vibrationaland torsional effects on the gearbox and engine. Further, at least onetorsional vibration dampener may be connected to the drive-train systemto dampen the harmonic effects of the reciprocating pump. According tosome embodiments, the at least one flywheel and the at least onetorsional damper may not require electrical control to be able tofunction, but it is contemplated that electrical sensors andinstrumentation may be present to monitor the condition of the driveline.

According to some embodiments, a pump system may include a pump, adriveshaft, driving equipment, and a vibration dampening assembly. Thepump may have an input shaft that is connected to the driveshaft. Thedriving equipment may include an output shaft that has an output flangeconnected to the driveshaft. The driving equipment may be configured torotate the driveshaft to rotate the input shaft of the pump therewith.The vibration dampening assembly may include at least one flywheel thatis operably connected to the input shaft and is configured to rotatetherewith. The input shaft may include an input flange that is connectedto the driveshaft. According to some embodiments, the at least oneflywheel may comprise a first flywheel.

According to some embodiments, the pump may be a single actingreciprocating pump. The first flywheel may be a single mass flywheel.The first flywheel may be connected to the output flange of the drivingequipment or the first flywheel may be connected to the input flange ofthe single acting reciprocating pump.

In some embodiments, the vibration dampening assembly may include atleast one torsional vibration damper that is operably connected to theinput shaft. According to some embodiments, the at least one torsionalvibration damper may comprise a first torsional vibration damper thatmay be connected to the input flange of the pump, may be connected tothe output flange of the driving equipment, and/or may be connected tothe first flywheel.

According to some embodiments, the first flywheel may be connected tothe output flange of the driving equipment and the first torsionalvibration damper may be connected to the first flywheel. The vibrationdampening assembly may include a second torsional vibration damper thatmay be connected to the input flange.

According to some embodiments, the vibration damping system may includea second flywheel that may be connected to the input flange. The secondtorsional vibration damper may be connected to the second flywheel.

According to some embodiments, the first and/or the second flywheel maybe configured to absorb a torque shock in the form of torque varianceresulting from hydraulic fluid pulsation within the pump. The firstand/or second torsional vibration damper may be configured to reducetorsional resonance within the driving equipment or the pump.

According to some embodiments, a method of sizing a flywheel for a pumpsystem that has a single acting reciprocating pump and driving equipmentconfigured to cycle the pump may include calculating a desired moment ofinertia of the flywheel and sizing the flywheel to have the desiredmoment of inertia. The desired moment of inertia may be calculated usinga kinetic energy “KE” of a torque variance within the pump system abovea nominal torque of the pump system that results from hydraulic fluidpulsation within the pump.

In some embodiments, calculating the desired moment of inertia of theflywheel may include calculating a first desired moment of inertia of afirst flywheel from a first portion of the kinetic energy “KE” of thetorque variance within the pump system as a result of hydraulic fluidpulsation within the pump, and calculating a second desired moment ofinertia of a second flywheel from a second portion of the kinetic energy“KE” of the torque variance within the pump system as a result ofhydraulic fluid pulsation within the pump. The first portion may begreater than, lesser than, or equal to the second portion. Sizing theflywheel may include sizing the first flywheel to have the first desiredmoment of inertia and sizing the second flywheel to have the seconddesired moment of inertia.

Still other aspects, embodiments, and advantages of these exemplaryaspects and embodiments, are discussed in detail below. Moreover, it isto be understood that both the foregoing information and the followingdetailed description are merely illustrative examples of various aspectsand embodiments, and are intended to provide an overview or frameworkfor understanding the nature and character of the claimed aspects andembodiments. Accordingly, these and other objects, along with advantagesand features of the present disclosure herein disclosed, will becomeapparent through reference to the following description and theaccompanying drawings. Furthermore, it is to be understood that thefeatures of the various embodiments described herein are not mutuallyexclusive and may exist in various combinations and permutations.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are included to provide a furtherunderstanding of the embodiments of the present disclosure, areincorporated in and constitute a part of this specification, illustrateembodiments of the present disclosure, and together with the detaileddescription, serve to explain the principles of the embodimentsdiscussed herein. No attempt is made to show structural details of thisdisclosure in more detail than may be necessary for a fundamentalunderstanding of the exemplary embodiments discussed herein and thevarious ways in which they may be practiced. According to commonpractice, the various features of the drawings discussed below are notnecessarily drawn to scale. Dimensions of various features and elementsin the drawings may be expanded or reduced to more clearly illustratethe embodiments of the disclosure.

FIG. 1 is a schematic view of a pump system having a first exemplaryembodiment of a vibration dampening assembly provided according to anembodiment of the disclosure.

FIG. 2 is a graph illustrating a pressure, acceleration, and suctionpressure of an exemplary pump of the pump system of FIG. 1 through acycle of the pump according to an embodiment of the disclosure.

FIG. 3 is a schematic front view of a flywheel of the pump system ofFIG. 1 according to an embodiment of the disclosure.

FIG. 4 is a schematic side view of the flywheel of the pump system ofFIG. 3 according to an embodiment of the disclosure.

FIG. 5 is a table providing exemplary properties of flywheels that eachhave the same moment of inertia.

FIG. 6 is another schematic front view of a flywheel of the pump systemof FIG. 1 illustrating bolt holes and rotational stresses of theflywheel according to an embodiment of the disclosure.

FIG. 7 is a graph illustrating tangential and radial stresses of theflywheel of FIG. 1 according to an embodiment of the disclosure.

FIG. 8 is a schematic side view of a portion of the pump system of FIG.1 illustrating a bolt and nut securing the flywheel to an output flangeaccording to an embodiment of the disclosure.

FIG. 9 is a schematic view of the pump system of FIG. 1 with anotherexemplary embodiment of a vibration dampening assembly according to anembodiment of the disclosure.

FIG. 10 is a schematic view of the pump system of FIG. 1 with anotherexemplary embodiment of a vibration dampening assembly according to anembodiment of the disclosure.

FIG. 11 is a schematic view of the pump system of FIG. 1 with anotherexemplary embodiment of a vibration dampening assembly according to anembodiment of the disclosure.

FIG. 12 is a graph showing torsional vibration analysis data resultsdemonstrating the reduction in synthesis and torque spikes with the useof a torsional vibration dampener (TVD) and a single mass produced by apump system such as shown in FIG. 1 according to an embodiment of thedisclosure.

DETAILED DESCRIPTION

The present disclosure will now be described more fully hereinafter withreference to example embodiments thereof with reference to the drawingsin which like reference numerals designate identical or correspondingelements in each of the several views. These example embodiments aredescribed so that this disclosure will be thorough and complete, andwill fully convey the scope of the disclosure to those skilled in theart. Features from one embodiment or aspect may be combined withfeatures from any other embodiment or aspect in any appropriatecombination. For example, any individual or collective features ofmethod aspects or embodiments may be applied to apparatus, product, orcomponent aspects or embodiments and vice versa. The disclosure may beembodied in many different forms and should not be construed as limitedto the embodiments set forth herein; rather, these embodiments areprovided so that this disclosure will satisfy applicable legalrequirements.

As used in the specification and the appended claims, the singular forms“a,” “an,” “the,” and the like include plural referents unless thecontext clearly dictates otherwise. In addition, while reference may bemade herein to quantitative measures, values, geometric relationships orthe like, unless otherwise stated, any one or more if not all of thesemay be absolute or approximate to account for acceptable variations thatmay occur, such as those due to manufacturing or engineering tolerancesor the like.

Referring now to FIG. 1, an exemplary pump system 1 having a vibrationdampening assembly 10 described in accordance with the presentdisclosure. The pump system 1 includes driving equipment 100 and drivencomponents including a driveshaft 200 and a pump 300. The vibrationdampening assembly 10 is secured to portions of a pump system 1 betweenthe driving equipment 100 and the pump 300 to dampen upstream highfrequency/low amplitude and low frequency/high amplitude torsionalvibrations generated by the operating pump 300 from reaching the drivingequipment 100.

The driving equipment 100 is illustrated as a power transfer case. Insome embodiments, the driving equipment 100 includes a driveshaft, atransmission, a gearbox, or an engine, e.g., an internal combustionengine or a gas turbine engine. The driving equipment 100 includes anoutput shaft 110 that has an output flange 112. The driving equipment100 is configured to rotate the output shaft 110 about a longitudinalaxis thereof. The driving equipment 100 may include an engine and atransmission, gearbox, and/or power transfer case that may be configuredto increase a torque and decrease a rotational speed of the output shaft110 relative to a driveshaft of the engine or that may be configured todecrease a torque and increase a rotational speed of the output shaft110 relative to a driveshaft of the engine. The pump 300 includes ininput shaft 310 having an input flange that is configure to receiveinput from the driving equipment 100 in the form of rotation of theinput flange about a longitudinal axis of the input shaft 310.

The driveshaft 200 has a driving or upstream portion 210, a driven ordownstream portion 240, and a central portion 230 between the upstreamand downstream portions 210, 240. The upstream portion 210 includes anupstream flange (not shown) that is connected to the output flange 112of the driving equipment 100 such that the upstream portion 210 rotatesin response or in concert with rotation of the output shaft 110. Thecentral portion 230 is secured to the upstream portion 210 and rotatesin concert therewith. The downstream portion 240 is secured to thecentral portion 230 and rotates in concert therewith. The downstreamportion 240 includes a downstream flange 242 that is connected to aninput flange of the pump 300 such that the input flange rotates inresponse or in concert with rotation of the driveshaft 200. Thedownstream portion 240 may also include a spindle 244 adjacent thedownstream flange 242. The upstream flange (not shown) may be similar todownstream flange 242 and the upstream portion 210 may include a spindle(not shown) that is similar to the spindle 244 of the downstream portion240.

In some embodiments, the output shaft 110 of the driving equipment 100is offset from the input shaft 310 of the pump 300 such that thelongitudinal axis of the output shaft 110 is out of alignment, i.e., notcoaxial with, the longitudinal axis of the input shaft 310. In suchembodiments, the upstream portion 210 or the downstream portion 240 mayinclude a constant velocity (CV) joint 220, 250 between the spindle 244and the central portion 230. The CV joints 220, 250 allow for the outputshaft 110 to be operably connected to the input shaft 310 when theoutput and input shafts 110, 310 are offset from one another.

During operation, the output shaft 110 is rotated by the drivingequipment 100 to rotate the input shaft 310 of the pump 300 such thatthe pump 300 is driven to pump slugs of fluid. Specifically, the drivingequipment 100 is configured to rotate the input shaft 310 at a constantvelocity such that the pump 300 provides a constant flow of fluid. Asthe pump 300 pumps slugs of fluid, the pulses of the slugs of fluidcreate a pulsation pressure that adds to the nominal operating pressureof the pump 300.

With additional reference to FIG. 2, the pressure P of the pump 300 isillustrated through an exemplary cycle of the pump 300. The pump 300 hasa nominal pressure P_(N) of 8250 psi with a normal operating pressure ina range of 7500 psi to 9000 psi. The pulsations of the operatingpressure illustrate the pulsation pressure described above which isknown as “hydraulic fluid pulsation.” This hydraulic fluid pulsation maylead to pressure spikes P_(S) as illustrated between points 60 and 150of the cycle of the pump 300 in FIG. 2. The pressure spikes P_(S) aremeasured as peak to peak pressure variations, which as shown in FIG. 2is 2,500 psi.

The hydraulic fluid pulsation describe above may be transferred upstreamfrom the pump 300 to the driving equipment 100 through the driveshaft200. Specifically, the hydraulic fluid pulsation results in torquevariations in a crank/pinion mechanism of the pump 300 that aretransferred upstream as torque output variations at the input shaft 310of the pump 300. These torque output variations may create a torsionalshock T_(S) at the output flange 112 of the output shaft 110. A singlelarge torsional shock T_(S) may damage components of the drivingequipment 100. In addition, an accumulation of minor or small torsionalshocks T_(S) may decrease a service life of one or more of thecomponents of the driving equipment 100.

With continued reference to FIG. 1, the vibration dampening assembly 10is provided to reduce the transfer of the torsional shock T_(S) upstreamto the driving equipment 100. The vibration dampening assembly 10 mayinclude at least one flywheel. In one aspect, the at least one flywheelmay comprise a flywheel 22 that is connected to the output flange 112and disposed about the upstream portion 210 of the driveshaft 200. Insome embodiments, the flywheel 22 may be connected to the output flange112 and be disposed about the output shaft 110.

As the output shaft 110 rotates the driveshaft 200, the flywheel 22rotates in concert with the output shaft 110. As shown in FIG. 3, torqueprovided by the driving equipment 100 to the input shaft 310 of the pump300 is illustrated as an input torque Ti and the torque outputvariations at the input shaft 310 of the pump 300 result in a reactiontorque illustrated as torque spikes T_(S). As the flywheel 22 rotates,angular momentum of the flywheel 22 counteracts a portion of or theentire torque output variances and reduces or eliminates torsional shockT_(S) from being transmitted upstream to the driving equipment 100.Incorporation of the flywheel 22 into the vibration dampening assembly10 allows for the vibration dampening assembly 10 to dampen the lowfrequency, high amplitude torsional vibrations imposed on the drivetrainsystem that is caused by the hydraulic fluid pulsation.

The angular momentum of the flywheel 22 may be calculated as arotational kinetic energy “KE” of the flywheel 22. The “KE” of theflywheel 22 may be used to absorb or eliminate a percentage of thetorsional shock T_(S). The “KE” of the flywheel 22 is a function of themoment of inertia “I” of the flywheel 22 and the angular velocity “ω” ofthe flywheel 22 which may be expressed as:

KE=½(Iω)²  (1)

As noted above, the driving equipment 100 is configured to rotate at aconstant angular velocity “ω” such that with a known “KE” or a knownmoment of inertia “I” the other of the “KE” or the moment of inertia “I”may be calculated. In addition, the moment of inertia “I” of theflywheel 22 is dependent on the mass “m” and the radial dimensions ofthe flywheel 22 and may be expressed as:

$\begin{matrix}{I = \frac{m\left( {{r_{1}}^{2} + {r_{2}}^{2}} \right)}{2}} & (2)\end{matrix}$

where r₁ is a radius of rotation and r₂ is a flywheel radius as shown inFIG. 3. This equation assumes that the flywheel 22 is formed of amaterial having a uniform distribution of mass. In some embodiments, theflywheel 22 may have a non-uniform distribution of mass where the massis concentrated away from the center of rotation to increase a moment ofinertia “I” of the flywheel 22 for a given mass. It will be appreciatedthat the mass may be varied for a given a radius of rotation r₁ and agiven a flywheel radius r₂ by varying a thickness “h” of the flywheel 22in a direction parallel an axis of rotation of the flywheel 22 as shownin FIG. 4.

The dimensions and mass of the flywheel 22 may be sized such that theflywheel 22 has a “KE” similar to a “KE” of an anticipated torquevariance above a nominal operating torque of the pump 300. In someembodiments, the flywheel 22 maybe sized such that the “KE” of theflywheel 22 is greater than an anticipated torque variance such that theflywheel has a “KE” greater than any anticipated torque variance and inother embodiments, the flywheel 22 may be sized such that the “KE” ofthe flywheel 22 is less than the anticipated torque variance such thatthe flywheel 22 is provided to absorb or negate only a portion of theanticipated torque variances. In particular embodiments, the flywheel 22is sized such that the “KE” of the flywheel 22 is equal to theanticipated torque variance such that the flywheel 22 is provided toabsorb or negate the anticipated torque variance while minimizing amoment of inertia “I” of the flywheel 22.

The rotational kinetic energy “KE” of the torque variance is calculatedfrom the specifications of a particular pump, e.g., pump 300, and fromempirical data taken from previous pump operations as shown in FIG. 2.For example, as shown in FIG. 2, the pressure spike P_(S) is analyzed todetermine a magnitude of the pressure spike P_(S) and a duration of thepressure spike P_(S). As shown, the duration of the pressure spike P_(S)occurred over 0.628 radians of the cycle and using the specification ofthe pump resulted in a torque above the nominal operating torque of 1420lb-ft. From these values and given the constant velocity of theparticular pump of 152.4 radians/second, the “KE” of a torque varianceresulting from the pressure spike P_(S) may be calculated as 8922 lb-ftor 12,097 N-m of work.

The “KE” of the torque variance may be used to size a flywheel 22 suchthat the flywheel 22 has a “KE” greater than or equal to the “KE” of thetorque variance. Initially, equation (1) is used to calculate a desiredmoment of inertia “I” of the flywheel 22 solving for the “KE” of thetorque variance created by the pressure spike P_(S) for a given angularvelocity “ω” of the flywheel 22. For example, the angular velocity “ω”of the output shaft 110 may be 152.4 radians/second with the “KE” of thetorque variance created by the pressure spike P_(S) being 12,097 N-m.Solving equation (1) provides a desired moment of inertia “I” of theflywheel 22 as 1.047 kg m².

Once the desired moment of inertia “I” of the flywheel 22 is determined,equation (2) is used to determine dimensions of the flywheel 22 usingdesired moment of inertia “I”. As shown in FIG. 4, with the desiredmoment of inertia “I”, a set radius of rotation “r₁”, and a setthickness of the flywheel 22, the flywheel radius “r₂” and mass “m” maybe manipulated such that the flywheel 22 has dimensions and a mass thatare optimized for a particular application. Referring to FIG. 4, forexample and not meant to be limiting, a 10 kg flywheel with an outerradius “r₂” of 0.45 m has the same moment of inertia as a 100 kgflywheel with an outer radius “r₂” of 0.13 m such that either the 10 kgflywheel or the 100 kg flywheel would have the same “KE” to absorb the“KE” of the torque variance created by the pressure spike P_(S).

It will be appreciated that for a given system, the radius of rotation“r₁” of the flywheel is set by a diameter of the spindle or flange onwhich the flywheel is secured, e.g., upstream flange of the upstreamportion 210 or the flange 242 or the spindle 244 of the downstreamportion 240 (FIG. 1). In addition, the thickness “h” of the flywheel 22may also be manipulated to vary a mass of the flywheel for a given outerradius “r₂”.

With additional reference to FIG. 6, the flywheel 22 is subjected torotational stresses that differ within the flywheel 22 dependent on theradial distance “r_(d)” away from axis of rotation “A_(R)” of theflywheel 22. It is important to choose a material for the flywheel 22that is capable of withstanding the rotational stresses of the flywheel22. To determine the rotational stresses of the flywheel 22, theflywheel may be treated as a thick-walled cylinder to calculate thetangential and radial stresses thereof. The calculations detailed belowassume that the flywheel 22 has a uniform thickness “h”, the flywheelradius “r₂” is substantially larger than the thickness “h” (e.g.,r₂>5h), and the stresses are constant over the thickness “h”. Thetangential stress “α_(t)” and radial stress “α_(r)” of the flywheel 22may be expressed as follows:

$\begin{matrix}{\sigma_{t} = {\rho{\omega^{2}\left( \frac{3 + v}{8} \right)}\left\{ {{r_{1}}^{2} + {r_{2}}^{2} + \frac{{r_{1}}^{2}\left( {r_{2}}^{2} \right)}{{r_{d}}^{2}} - {\frac{\left( {1 + {3v}} \right)}{3 + v}\left( {r_{d}}^{2} \right)}} \right\}}} & (3) \\{\sigma_{r} = {\rho{\omega^{2}\left( \frac{3 + v}{8} \right)}\left\{ {{r_{1}}^{2} + {r_{2}}^{2} - \frac{{r_{1}}^{2}\left( {r_{2}}^{2} \right)}{{r_{d}}^{2}} - \left( {r_{d}}^{2} \right)} \right\}}} & (4)\end{matrix}$

where ρ is a mass density (lb./in³) of the material of the flywheel 22,ω is the angular velocity (rad/s) of the flywheel 22, and v is thePoisson's ratio of the flywheel 22. As shown in FIG. 7, when the innerradius r₁ is 2.5 inches and the outer radius r₂ is 8.52 inches themaximum tangential stress “α_(t)” is 1027 psi at 2.5 inches from theaxis of rotation and the maximum radial stress “α_(r)” is 255 psi at 4.5inches from the axis of rotation.

The installation or securement of the flywheel 22 to the pump system,e.g., to output flange 112 of the output shaft 110 (FIG. 1), must alsobe analyzed to confirm that the means for attachment is suitable for thecalculated stresses. For example, the planar stresses occurring at thepoint of installment may be calculated. Specifically, the flywheel 22may be installed to the output flange 112 as described above or to theinput flange of the pump as described below. For the purposes of thisanalysis, it will be assumed that the flywheel 22 is installed with anumber of bolts 72 and nuts 76 as shown in FIG. 8. To secure theflywheel 22 to the output flange 112 (FIG. 1), each bolt 72 is passedthrough a bolt hole 70 defined through the flywheel 22 at a bolt radius“r_(B)” (FIG. 6) from the axis of rotation “A_(R)” of the flywheel 22.The planar stresses may be calculated as follows:

$\begin{matrix}{F_{B} = \frac{T}{r_{B}}} & (5) \\{v_{s} = \frac{T}{A_{B}}} & (6) \\{v_{b} = \frac{F_{B}}{hd}} & (7)\end{matrix}$

where F_(B) is a force (lbf) applied to the bolt 72, T is a torque(lb-ft) applied to the flywheel 22, A_(B) is a bolt bearing stress area(in²) of the bolt 72, d is a diameter (ft) of the bolt hole 70, vs is ashear stress (psi) of each bolt 72, and v_(b) is a bearing stress on theflywheel 22/bolt hole 70 (psi).

Continuing the example above, given a maximum torque “T” applied to theoutput flange 112 of 35,750 lb-ft with a bolt radius “r_(B)” of 7.6inches, the force applied to the bolts F_(B) is 56,447 lbf. With thebolt bearing area of each bolt 72 being 0.785 in² the shear stress vs ofeach of the 10 bolts is 7,187 psi. With the thickness of the flywheel“h” being 1.54 inches and a diameter of each bolt hole being 1.06inches, the bearing stress v_(B) is 3,885 psi.

From the calculated stresses of the example above and applying a factorof safety, a material for the flywheel 22 should have should have atensile yield strength greater than or equal to 75 ksi. Examples of somesuitable materials for the flywheel 22 are 1040 carbon steel, 1050carbon steel, or Inconel® 718; however, other suitable metals or othermaterials may also be used. In addition, the materials sued for thebolts 72 and the nuts 76 should have a tensile strength greater than thecalculated stresses. Examples of some suitable materials for the bolts72 and the nuts 76 are Grade 8 carbon steel, Grade 5 carbon steel, orGrade G (8) steel; however, other suitable metals or other materials mayalso be used.

Referring briefly back to FIG. 1, the vibration dampening assembly 10may also include at least one torsional vibration damper. The at leastone torsional vibration damper may comprise a torsional vibration damper24 disposed upstream of the pump 300. As shown, the torsional vibrationdamper 24 is disposed about the upstream portion 210 of the driveshaft210 and is connected to a downstream side of the flywheel 22. Thevibration damper 24 may be connected directly to the flywheel 22 ordirectly to the output flange 112 of the driving equipment 100 and maybe disposed about the upstream portion 210 of the driveshaft 210 or theoutput shaft 110. The torsional vibration damper 24 is configured toprevent torsional resonance within the driving equipment 100 that maylead to damage or fatigue of components of the driving equipment 100,the driveshaft 200, or the pump 300. Incorporation of the torsionalvibration damper 24 along the drivetrain in between the gearbox and/ortransmission and the single acting reciprocating pump 300 allows for thevibration dampening assembly 10 to dampen the high frequency, lowamplitude torsional vibrations imposed on the drivetrain system that iscaused by forced excitations from the synchronous machinery. Thetorsional vibration damper 24 may be a viscous, a spring-viscous, or aspring torsional vibration damper. Examples of suitable torsionalvibration dampers include, but are not limited to, a Geislinger Damper,a Geislinger Vdamp®, a Metaldyne Viscous Damper, a Kendrion TorsionalVibration Dampener, a Riverhawk Torsional Vibration Dampener, and thelike.

As shown FIG. 1, the vibration dampening assembly 10 is secured to theoutput flange 112. Specifically, the flywheel 22 is connected to theoutput flange 112 and the torsional vibration damper 24 is connected tothe flywheel 22. However, as illustrated below with reference to FIGS.5-7, the flywheel 22 and/or the torsional vibration damper 24 may bedisposed at other positions within the pump system 1 and the vibrationdampening assembly 10 may include multiple flywheels and/or multiplevibration dampers.

Referring now to FIG. 9, the vibration dampening assembly 10 includes afirst flywheel 22, the torsional vibration damper 24, and a secondflywheel 32. The second flywheel 32 is connected to the input flange ofthe pump 300. When the vibration dampening assembly 10 includes thefirst flywheel 22 and the second flywheel 32, the sum of the “KE” of theflywheels 22, 32 may be configured in a manner similar to the “KE” of asingle flywheel as detailed above with respect to the flywheel 22. Insome embodiments, each of the first and second flywheel 22, 32 is sizedto have a similar moment of inertia “I”. In such embodiments, the firstand second flywheel 22, 32 may have similar dimensions and mass or mayhave different dimensions and mass while having a similar moment ofinertia “I”. In other embodiments, the first flywheel 22 is configuredto have a moment of inertia “I” different, e.g., greater than or lesserthan, a moment of inertia “I” of the second flywheel 32.

With reference to FIG. 10, the vibration dampening assembly 10 includesthe flywheel 22, a first torsional vibration damper 24, and a secondvibration damper 34. The flywheel 22 is connected to the output flange112 of the driving equipment 100 and the first torsional vibrationdamper 24 is connected to the flywheel 22. The second vibration damper34 is connected to the input flange of the pump 300. Using first andsecond vibration dampers 24, 34 instead of a single vibration damper mayallow for greater resistance to torsional resonance within the drivingequipment 100 and/or for each of the first and second vibration dampers24, 34 to have a reduced size compared to a single vibration damper.

Referring now to FIG. 11, the vibration dampening assembly 10 includesthe first flywheel 22, the first torsional vibration damper 24, thesecond flywheel 32, and the second vibration damper 34. The firstflywheel 22 is connected to the output flange 122 of the drivingequipment 100 with the first torsional vibration damper 24 connected tothe first flywheel 22. The second flywheel 32 is connected to the inputflange of the pump 300 with the second torsional vibration damper 34connected to the second flywheel 32. As noted above, the first andsecond flywheels 22, 32 may be sized such that the sum of the “KE” ofthe flywheels 22, 32 is configured in a manner similar to the “KE” of asingle flywheel detailed above with respect to the flywheel 22. Inaddition, using first and second vibration dampers 24, 34 instead of asingle vibration damper which may allow for greater resistance totorsional resonance within the driving equipment 100.

The configurations of the vibration dampening assembly 10 detailed aboveshould be seen as exemplary and not exhaustive of all the configurationsof the vibration dampening assembly 10. For example, the vibrationdampening assembly 10 may consist of a flywheel 32 and a torsionalvibration damper 34 as shown in FIG. 6. In addition, it is contemplatedthat the vibration dampening assembly 10 may include more than twoflywheels or more than two torsional vibration dampers. Further, thevibration dampers may each be connected directly to a respective flange,e.g., output flange 112 or input flange, and not be directly connectedto a flywheel, e.g., flywheels 22, 32.

FIG. 12 is a graph showing torsional vibration analysis data resultsdemonstrating the reduction in synthesis and torque spikes with the useof a torsional vibration dampener (TVD) and a single mass produced by apump system such as shown in FIG. 1 according to an embodiment of thedisclosure. A significant reduction in amplitude and frequency of thesystem torque spikes is noticeable over entire speed range of thereciprocating pump.

This is a continuation of U.S. Non-Provisional application Ser. No.17/363,151, filed Jun. 30, 2021, titled “SYSTEMS AND METHOD FOR USE OFSINGLE MASS FLYWHEEL ALONGSIDE TORSIONAL VIBRATION DAMPER ASSEMBLY FORSINGLE ACTING RECIPROCATING PUMP,” which is a continuation of U.S.Non-Provisional application Ser. No. 17/213,562, filed Mar. 26, 2021,titled “SYSTEMS AND METHOD FOR USE OF SINGLE MASS FLYWHEEL ALONGSIDETORSIONAL VIBRATION DAMPER ASSEMBLY FOR SINGLE ACTING RECIPROCATINGPUMP,” now U.S. Pat. No. 11,092,152, issued Aug. 17, 2021, which is acontinuation of U.S. Non-Provisional application Ser. No. 16/948,291,filed Sep. 11, 2020, titled “SYSTEMS AND METHOD FOR USE OF SINGLE MASSFLYWHEEL ALONGSIDE TORSIONAL VIBRATION DAMPER ASSEMBLY FOR SINGLE ACTINGRECIPROCATING PUMP,” now U.S. Pat. No. 11,015,594, issued May 25, 2021,which claims priority to and the benefit of U.S. Provisional ApplicationNo. 62/704,560, filed May 15, 2020, titled “SYSTEMS AND METHOD FOR USEOF SINGLE MASS FLYWHEEL ALONGSIDE TORSIONAL VIBRATION DAMPER ASSEMBLYFOR SINGLE ACTING RECIPROCATING PUMP,” and U.S. Provisional ApplicationNo. 62/899,963, filed Sep. 13, 2019, titled “USE OF SINGLE MASS FLYWHEELALONGSIDE TORSIONAL VIBRATION DAMPER SYSTEM FOR SINGLE ACTINGRECIPROCATING PUMP,” the disclosures of which are incorporated herein byreference in their entireties.

While several embodiments of the disclosure have been shown in thedrawings, it is not intended that the disclosure be limited thereto, asit is intended that the disclosure be as broad in scope as the art willallow and that the specification be read likewise. Any combination ofthe above embodiments is also envisioned and is within the scope of theappended claims. Therefore, the above description should not beconstrued as limiting, but merely as exemplifications of particularembodiments. Those skilled in the art will envision other modificationswithin the scope of the claims appended hereto.

What is claimed:
 1. A method of using a vibration dampening assembly fora pump system, the assembly comprising: positioning one or moretorsional vibration dampers operably to connect to an input drive shaftof the pump system so as to reduce torsional resonance within one ormore of: (a) driving equipment for one or more pumps of the pump system,or (b) the one or more pumps of the pump system, the one or moretorsional vibration dampers including a first torsional vibration damperoperably to connect to an output drive shaft and a second torsionalvibration damper operably to connect to the input drive shaft; androtating one or more flywheels, including a first flywheel operablyconnected to the input drive shaft of the pump system and configured torotate with the input drive shaft and connected to the first torsionalvibration damper, thereby to absorb a torque shock in the form of torquevariance within the one or more pumps of the pump system.
 2. The methodas defined in claim 1, wherein the one or more pumps includes a singleacting reciprocating pump, and wherein the first flywheel comprises asingle mass flywheel.
 3. The method as defined in claim 1, wherein thefirst torsional vibration damper is configured to connect to the outputdrive shaft.
 4. The vibration dampening assembly as defined in claim 1,wherein the first flywheel is to configured operably to connect to theoutput drive shaft.
 5. The vibration dampening assembly as defined inclaim 1, wherein the one or more flywheels further comprises a secondflywheel, the second flywheel being configured to connect to the inputdrive shaft.
 6. A method of manufacturing a single mass flywheel for avibration assembly of a pump system having one or more reciprocatingpumps and associated driving equipment to drive the one or more pumps,the method comprising: determining a desired moment of inertia of theflywheel by a controller from kinetic energy of a torque variance withinthe pump system above a nominal torque of the pump system resulting fromhydraulic fluid pulsation within the one or more pumps, the determiningof the desired moment of inertia of the flywheel including: determininga first desired moment of inertia of a first flywheel from a firstportion of the kinetic energy of the torque variance within the pumpsystem resulting from fluid pulsation within the one or more pumps; anddetermining a second desired moment of inertia of a second flywheel froma second portion of the kinetic energy of the torque variance within thepump system resulting from fluid pulsation within the one or more pumps;determining, by the controller, one or more of a first flywheelrotational stress associated with the first flywheel or a secondflywheel rotational stress associated with the second flywheel, the oneor more of the first flywheel rotational stress or the second flywheelrotational stress includes one or more of: (a) a first radial stress anda first tangential stress associated with the first flywheel; or (b) asecond radial stress and a second tangential stress associated with thesecond flywheel; sizing the flywheel to have the desired moment ofinertia from the determined moment of inertia and the determinedrotational stress, the sizing the flywheel including sizing the firstflywheel to have the first desired moment of inertia and sizing thesecond flywheel to have the second desired moment of inertia; andproducing the flywheel for the pump system based on the sizing of theflywheel.
 7. The method as defined in claim 6, wherein sizing theflywheel comprises one or more of: determining a first mass of the firstflywheel based at least in part on the first desired moment of inertiaby the controller, or determining a second mass of the second flywheelbased at least in part on the second desired moment of inertia by thecontroller; and wherein the first portion of the kinetic energy beinggreater than, lesser than, or equal to the second portion.
 8. The methodas defined in claim 6, further comprising determining, by the controllerand based at least in part on one or more of the first mass of the firstflywheel or the second mass of the second flywheel, one or more of: oneor more of a first radius of rotation of the first flywheel or a firstthickness of the first flywheel; or one or more of a second radius ofrotation of the second flywheel or a second thickness of the secondflywheel.
 9. The method as defined in claim 6, further comprisingdetermining, by the controller, one or more of: the first portion of thekinetic energy of the torque variance within the pump system; or thesecond portion of the kinetic energy of the torque variance within thepump system.
 10. The method as defined in claim 9, wherein the one ormore of the first portion of the kinetic energy of the torque variancewithin the pump system or the second portion of the kinetic energy ofthe torque variance is determined based at least in part on empiricaldata associated with previous operations of the pump.
 11. The method asdefined in claim 10, wherein the empirical data comprises one or more ofa magnitude of pressure spikes or a duration of pressure spikesoccurring during the previous operations of the pump.
 12. The method asdefined in claim 16, wherein one or more of the first portion of thekinetic energy of the torque variance within the pump system or thesecond portion of the kinetic energy of the torque variance isdetermined based at least in part on one or more of a first angularvelocity of operation of the pump or a second angular velocity ofoperation of the pump.
 13. A method of manufacturing a single massflywheel for a vibration assembly of a pump system having one or morereciprocating pumps and associated driving equipment to drive the one ormore pumps, the method comprising: determining a desired moment ofinertia of the flywheel by a controller from kinetic energy of a torquevariance within the pump system above a nominal torque of the pumpsystem resulting from hydraulic fluid pulsation within the one or morepumps, the determining of the desired moment of inertia of the flywheelincluding: determining a first desired moment of inertia of a firstflywheel from a first portion of the kinetic energy of the torquevariance within the pump system resulting from fluid pulsation withinthe one or more pumps; and determining a second desired moment ofinertia of a second flywheel from a second portion of the kinetic energyof the torque variance within the pump system resulting from fluidpulsation within the one or more pumps; determining, by the controller,one or more of a first flywheel rotational stress associated with thefirst flywheel or a second flywheel rotational stress associated withthe second flywheel, the one or more of the first flywheel rotationalstress or the second flywheel rotational stress includes one or more of:(a) a first radial stress and a first tangential stress associated withthe first flywheel; or (b) a second radial stress and a secondtangential stress associated with the second flywheel; sizing theflywheel to have the desired moment of inertia from the determinedmoment of inertia, the sizing the flywheel including sizing the firstflywheel to have the first desired moment of inertia and sizing thesecond flywheel to have the second desired moment of inertia; andproducing the flywheel for the pump system based on the sizing of theflywheel.
 14. The method as defined in claim 13, wherein sizing theflywheel comprises one or more of: determining a first mass of the firstflywheel based at least in part on the first desired moment of inertiaby the controller, or determining a second mass of the second flywheelbased at least in part on the second desired moment of inertia by thecontroller.
 15. The method as defined in claim 13, further comprisingdetermining, by the controller and based at least in part on one or moreof the first mass of the first flywheel or the second mass of the secondflywheel, one or more of: one or more of a first radius of rotation ofthe first flywheel or a first thickness of the first flywheel; or one ormore of a second radius of rotation of the second flywheel or a secondthickness of the second flywheel.
 16. The method as defined in claim 13,further comprising determining, by the controller, one or more of: thefirst portion of the kinetic energy of the torque variance within thepump system; or the second portion of the kinetic energy of the torquevariance within the pump system.
 17. The method as defined in claim 13,wherein the one or more of the first portion of the kinetic energy ofthe torque variance within the pump system or the second portion of thekinetic energy of the torque variance is determined based at least inpart on empirical data associated with previous operations of the pump.18. The method as defined in claim 17, wherein the empirical datacomprises one or more of a magnitude of pressure spikes or a duration ofpressure spikes occurring during the previous operations of the pump.19. The method as defined in claim 18, wherein one or more of the firstportion of the kinetic energy of the torque variance within the pumpsystem or the second portion of the kinetic energy of the torquevariance is determined based at least in part on one or more of a firstangular velocity of operation of the pump or a second angular velocityof operation of the pump.
 20. A method of manufacturing a single massflywheel for a vibration assembly of a pump system having one or morereciprocating pumps and associated driving equipment to drive the one ormore pumps, the method comprising: determining a desired moment ofinertia of the flywheel by a controller from kinetic energy of a torquevariance within the pump system above a nominal torque of the pumpsystem resulting from hydraulic fluid pulsation within the one or morepumps, the determining of the desired moment of inertia of the flywheelincluding: determining a first desired moment of inertia of a firstflywheel from a first portion of the kinetic energy of the torquevariance within the pump system resulting from fluid pulsation withinthe one or more pumps; and determining a second desired moment ofinertia of a second flywheel from a second portion of the kinetic energyof the torque variance within the pump system resulting from fluidpulsation within the one or more pumps; sizing the flywheel to have thedesired moment of inertia from the determined moment of inertia, thesizing the flywheel including sizing the first flywheel to have thefirst desired moment of inertia and sizing the second flywheel to havethe second desired moment of inertia; and producing the flywheel for thepump system based on the sizing of the flywheel.
 21. The method asdefined in claim 20, wherein sizing the flywheel comprises one or moreof: determining a first mass of the first flywheel based at least inpart on the first desired moment of inertia by the controller, ordetermining a second mass of the second flywheel based at least in parton the second desired moment of inertia by the controller.
 22. Themethod as defined in claim 20, further comprising determining, by thecontroller and based at least in part on one or more of the first massof the first flywheel or the second mass of the second flywheel, one ormore of: one or more of a first radius of rotation of the first flywheelor a first thickness of the first flywheel; or one or more of a secondradius of rotation of the second flywheel or a second thickness of thesecond flywheel.
 23. The method as defined in claim 20, furthercomprising determining, by the controller, one or more of: the firstportion of the kinetic energy of the torque variance within the pumpsystem; or the second portion of the kinetic energy of the torquevariance within the pump system.
 24. The method as defined in claim 20,wherein the one or more of the first portion of the kinetic energy ofthe torque variance within the pump system or the second portion of thekinetic energy of the torque variance is determined based at least inpart on empirical data associated with previous operations of the pump.25. The method as defined in claim 24, wherein the empirical datacomprises one or more of a magnitude of pressure spikes or a duration ofpressure spikes occurring during the previous operations of the pump.26. The method as defined in claim 25, wherein one or more of the firstportion of the kinetic energy of the torque variance within the pumpsystem or the second portion of the kinetic energy of the torquevariance is determined based at least in part on one or more of a firstangular velocity of operation of the pump or a second angular velocityof operation of the pump.
 27. The method as defined in claim 26, furthercomprising determining, by the controller, one or more of a firstflywheel rotational stress associated with the first flywheel or asecond flywheel rotational stress associated with the second flywheel,the one or more of the first flywheel rotational stress or the secondflywheel rotational stress includes one or more of: (a) a first radialstress and a first tangential stress associated with the first flywheel;or (b) a second radial stress and a second tangential stress associatedwith the second flywheel.